A study on cayley fuzzy graphs and cayley fuzzy graph structures

Abstract

As most of real life problems deals with vague or imprecise concepts, the theory of fuzzy sets has a wider application than that of classical theories. The fuzzy set is a generalisation of fundamental mathematical concept of a set. Most of the mathematical theories can be extended using the concepts of a fuzzy set and fuzzy logic. A few among the real world problems where this theory has application are pattern recognition, information processing, increasing the efficiency of a system and multivalued decision processing. Similarly the theory of graphs is an extremely useful tool for solving combinatorial problems in different areas such as geometry, algebra, number theory, topology, operation research, optimization, and computer science. This thesis mainly deals with the study of Cayley fuzzy graphs and digraph structures induced by some algebraic structures. Many graph properties are expressed in terms of algebraic properties. Main focus is on the study of Cayley fuzzy graphs, Cayley bipolar fuzzy graphs and Cayley intuitionistic fuzzy graphs induced by loops, a weaker structure than groups. Moreover, we study Cayley fuzzy digraph structure and Cayley bipolar fuzzy digraph structure induced by groups and loops. The concepts of Cayley fuzzy graphs were first introduced and studied by Namboothiri et. al.. A nat- ural question is: Does weaker algebraic structure induce Cayley fuzzy graphs? Here in this thesis we prove that the algebraic structure loop induces Cayley fuzzy/bipolar fuzzy graphs and structures.This thesis comprises of six chapters. The first chapter contains the preliminary definitions and results that were used in the remaining chapters. In second chapter, we introduced the concept of Cayley fuzzy graphs induced by loops and studied graph theoretic properties in terms of algebraic properties. In the third chapter we introduced Cayley bipolar fuzzy graphs induced by loops and studied many basic properties. In fourth chapter Cayley intuitionistic fuzzy graph is defined. In the next two chapters, fifth and sixth, we extend our studies to digraph structures induced by groups and also to those induced by loops.